Class 5
\(\displaystyle\ln' x = \frac{1}{x}\)
\(\exp'(x) = \exp(x)\)
\(\displaystyle\arcsin' y = \frac{1}{\sqrt{1 - y^2}}\)
\(\displaystyle\arccos' x = -\frac{1}{\sqrt{1 - x^2}}\)
\(\displaystyle\arctan' z = \frac{1}{1 + z^2}\)
\(\displaystyle\operatorname{arccot}' w = -\frac{1}{1 + w^2}\)
\(\displaystyle \int \frac{1}{x}\;dx = \begin{cases} \ln(-x) + C_1 & \text{if } x < 0\\ \ln x + C_2 & \text{if } x > 0\end{cases}\)
\(\displaystyle \int \exp(x)\;dx = \exp(x) + C\)
\(\displaystyle \int \frac{1}{\sqrt{1 - y^2}}\;dy = \arcsin y + C_1 = -\arccos y + C_2\)
\(\displaystyle \int \frac{1}{1 + z^2}\;dz = \arctan z + C_1 = -\operatorname{arccot} z + C_2\)
Find \(f'(x)\) if \(f(x) = \ln \sin(x)\).
Find the maxima, minima, inflection points, asymptotes, intervals of increase and decrease etc. of the function \(g(x) = \exp(-x^2)\).
Let \(f(x) = 2^x\). Find \(f'(x)\).
Find \(g'(x)\) if \(g(x) = \arctan(2x + 1)\).
\(\displaystyle \int_0^1 \frac{1}{2x + 3}\;dx\)
\(\displaystyle \int_0^1 \frac{x}{x^2 + 4}\;dx\)
\(\displaystyle \int_0^1 \frac{1}{x^2 + 4}\;dx\)
\(\displaystyle \int_3^5 \frac{1}{x^2 - 6x + 13}\;dx\)
\(\displaystyle \int_0^{\pi/4} \tan t\;dt\)
\(\displaystyle \int_0^1 x\exp(-x^2)\;dx\)
\(\displaystyle \int_0^1 \frac{e^x}{e^x + 1}\;dx\)
\(\displaystyle \int_0^1 \frac{e^x + 1}{e^x}\;dx\)
Hyperbolic cosine: \[\cosh t = \frac{e^t + e^{-t}}{2}\]
Hyperbolic sine: \[\sinh t = \frac{e^t - e^{-t}}{2}\]
Hyperbolic tangent: \[\tanh t = \frac{\sinh t}{\cosh t} = \frac{e^t - e^{-1}}{e^t + e^{-t}}\]